Group Theoretical Approach in Using Canonical Transformations and Symplectic Geometry in the Control of Approximately Modelled Mechanical Systems Interacting with Unmodelled Environment
J.K. Tar*, I.J. Rudas*, J.F. Bitó** *
Department of Information Technology, **Centre of Robotics and Automation Bánki Donát Polytechnic H-1081 Budapest, Népszínház u. 8., Hungary Fax: +36-1-133-9183, E-mail:
[email protected],
[email protected],
[email protected]
Abstract In spite of its simpler structure than that of the Euler-Lagrange equations-based model, Hamiltonian formulation of Classical Mechanics (CM) gained only limited application in the Computed Torque Control (CTC) of the rather conventional robots. A possible reason for this situation may be, that while the independent variables of the Lagrangian model are directly measurable by common industrial sensors and encoders, the Hamiltonian canonical coordinates are not measurable and can also not be computed in the lack of detailed information on the dynamics of the system under control. As a consequence, transparent and lucid mathematical methods bound to the Hamiltonian model utilizing the special properties of such concepts as Canonical Transformations, Symplectic Geometry, Symplectic Group, Symplectizing Algorithm, etc. remain out of the reach of Dynamic Control approaches based on the Lagrangian model. In this paper the preliminary results of certain recent investigations aiming at the introduction of these methods in dynamic control are summarized and illustrated by simulation results. The propoesed application of the Hamiltonian model makes it possible to achieve a rigorous deductive analytical treatment up to a well defined point exaclty valid for a quite wide range of many different mechanical systems. From this point on it reveals such an ample assortment of possible non-deductive, intuitive developments and
approaches even within the investigations aiming a particular paradigm that publication of these very preliminary and early results seems to have definite reason, too. 1. Introduction In order to gain precise quantitative description of the physical processes different physical quantities and concepts must be provided with some real numbers or certain groups of real numbers. In general the process of this provision is realized by the aid of different measurements. Due to the objective nature of the measuring process, especially in the field of technical applications, the illusion that fully objective meaning can be attributed to the numbers being results of the measurements frequently arises. Though this attitude also is supported by practical considerations concerning the direct measurability of certain quantities, it is misleading in the sense that this "provision" (that is the measurement) has many arbitrary possibilities and that physical concepts can be modelled mathematically in a higher level of abstraction. A particular field of significant practical interest is Classical Mechanics since in our daily life we meet many equipment for the behavior of which partly the laws of CM are responsible. Typical examples are industrial robots as non-linear, strongly coupled multiple variable systems for the fault-tolerant control of which many recent efforts were exerted (e.g. Tosunoglu1,2). In the field of CM, for instance, the basic concept is the set of possible physical states of the system forming a differentiable manifold. For gaining quantitative description, differentiable manifolds can be described by the use of atlases consisting of contradiction-free maps mapping some subsets of the manifold to some open regions of n. The coordinates of these maps are not necessarily the direct results of certain measurements: they may and must be related to the measurements in indirect ways. By introducing topology-conserving coordinate transformations defined over the coordinates of a given map, new maps can be introduced for dealing with the same physical reality.
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It is evident, that the mathematical form and complexity of the equations describing the same physical process may considerably depend on the properties of the given map which from this point on will be referred to as a particular representation. In the control of mechanical devices as industrial robots and manipulators the practical need for direct measurability of the coordinates to be controlled results in a strong insistence on the Lagrangian model's generalized coordinates. The use of the Lagrangian model may command the high price that the appropriate equations of motion form a second-order nonlinear set of differential equations. The dynamic constants (parameters) of the robot as mechanical system in this model are "hidden" in the elements of a positive definite symmetric quadratic matrix referred to as the inertia matrix of the robot. The same parameters also are present in the quadratic expressions regarding the first time derivatives of the generalized coordinates, as well as in the terms expressing the effect of the gravitation. This second order nature, at least within the frames of the Lagrangian model excludes the possibility of using simple geometric concepts in desribing the propagation of the state of the system in time. For this purpose first order equations are needed in which the first time derivatives of the coordinates describing the state of the system can be interpreted as the elements of the tangent space of the differentiable manifold of the states. As is well known, within the frames of CM this step first was made by Hamilton in the 19th century by introducing the so called canonical coordinates as the results of a possible Legendre-transformation. Regarding the mathematical structure of CM this step had far reaching consequences. The theory gained the possibility of having a pure local geometric interpretation leading to the concept of Symplectic Geometry defined on the tangent space of the states. Symplectic Geometry has considerable formal analogies with the properties of the Euclidean Geometry more familiar in our everiday-life. Both concepts are based on a basic quadratic structure referred to as the scalar product and the symplectic structure, respectively. On the basis of these concepts the sets of the orthonormal and symplectic sets of linearly independent basis vectors can be introduced.
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In Hamiltonian Mechanics the use of symplectic sets of basis vectors instead of orthonormal ones has definite reason: the state propagation of the mechanical systems transforms symplectic sets into symplectic ones in the tangent space of the physical states. Therefore, the description of the mechanical systems by symplectic sets has a kind of tranasparent "symmetry" which is not "apparent" in the case of other representations. As in the case of the Euclidean Geometry in which an orthonormal set of basis vectors can be chosen in many arbirtary ways, in Symplectic Geometry also many arbitrary possibilities are available for choosing some symplectic basis. From the point of view of algorithmic considerations, in both cases an appropriate number of arbitrary but linearly independent vectors can be chosen in the first step. The free parameters of the arbitrary possible choices for the orthonormed (symplectic) basis are "hidden" in these "initial" vectors. By the use of simple and easily programmable numerical algorithms (the Gram-Schmidt and the Symplectising one) appropriate orthonormal (symplectic) basis can be gained from the initial vectors. Group Theory-based analysis of the free parameters in the appropriate cases leads to the concepts of the orhogonal and the symplectic groups, respectively. From physical point of view, as the orthogonal group describes an inner symmetry of Newton's "absolute space" observable by our senses, the symplectic group expresses an "abstract", non-trivial inner symmetry
of the conservative mechanical systems. Both groups consist of unimodular
matrices which can be inverted by simple matrix multiplications requiring very limited number of numerical operations. Furthermore, both groups are Lie-groups and can be parametrized in many arbitrary ways with continuous parameters making it possible to use closed analytical formulas for describing the appropriate elements of the groups. In this description the linearly independent vectors of the tangent space of the groups near the vicinity of the unity element (the so-called generators) play key role. By using the Liealgebras of the appropriate groups, besides the geometric ones considerable algebraic analogies can be utilized, too. In the Hamiltonian model the propagation of the state of the
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physical system simply can be related to the gradient of a scalar function, the Hamiltonian of the conservative mechanical system. In spite of the above listed formal advantages, the Hamiltonian model has a serious drawback: its canonical coordinates cannot be directly measured. They can be calculated from the directly measurable coordinates only in the case of exactly knowing the dynamical parameters of the system normally not available in the case of a given robot. Besides being complicated and time-consuming processes, these calculations cannot be done without identifying the unknown parametres of the system. As it was shown by Lantos 3 in 1993, normally only different combinations of the dynamical parameters of the robots can be identified via long lasting off-line calculations. (The appropriate "groups" depend on the kinematic strucure of the robot arms, too.) The results of these calculations also are influenced by the dynamic interaction between the robot arm and its environment. On the basis of the above calculations it is clear, that in order to utilize the formal advantages of Hamiltonian Mechanics invention of some "additional idea" is necessary. The "progenitor" of this idea first was introduced by Jánossy4 in a quite different context. In his cited work Jánossy made an attempt to generalize Einstein's Special Theory of Relativity via introducing a so called "Deformation Principle". The essence of this idea is to use two different interpretations of the Lorentz Group+ . In the "conventional interpretation" a Lorentz transformation yields the description of the same physical system by the use of the coordinates of a "system of coordinates" or frame different to the original one. Jánossy's interpretation runs as follows. By using the coordinates of a well defined frame, by transforming the coordinates of a given physical system with a Lorentz transformation the coordinates of a different possible physical system, the deformed one can be gained. For maintaining the possibility of this interpretation introduction of certain restrictions reagrding the allowable transformations was necessary. Based on this analogy, a similar "deformation +
It is worthy of note, that the Special Theory of Relativity also has quite strict formal analogies with both the Symplectic and the Euclidean Geometries. The basic concept is a quadratic term describing the propagation of light signals. This basic concept leads to the so called Minkowski Geometry and to the Lorentz Group in quite similar manner as the Orthogonal Group is related to the Euclidean, and the Symplectic Group is pertaining to the Symplectic Geometry, respectively.
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principle" was proposed by Tar et al5 for the interpretation of the elements of the Symplectic Group strictly related to the Canonical Transformations of the Hamiltonian Mechanics. The main idea was based on a very rough dynamic model of the robot to be controlled. In this model the appropriate local symplectic deformations were interpreted as adaptive modifications of the initial rough model. Due to giving up the demand of full identification of the system, the appropriate procedure also was referred to as "partial identification". By using a symple paradigm for numerical simulations some results of preliminary investigations considering the operation of the outlined method were announced in different international conferences. These publications utilized only certain parts of the possibilities mentioned in this introduction. Recent investigations revealed, that on the basis of this "deformation principle" many different, adaptive control ideas can be developed. Besides being strictly related to the mathematical structure of the Hamiltonian Mechanics, these methods also show strong similarities with modern computing technologies as Soft Computing. In both cases the identification and use of the exact dynamic model of the system to be controlled is given up. Either the Fuzzy Systems (FS), or the Artificial Neural Network (ANN) have well defined mathematical structure in which a plenty of free parameters are "hidden". The learning process of ANNs essentially consits of some tuning of these free parameters. In the case of supervised training (either purely causal, stochastic or combined) simple rules can be applied for realizing this learning. Fuzzy systems are also based on a plenty of free parameters hidden in the shape of the "membership functions" and "fuzzy relations". There exist tunable "adaptive" fuzzy systems, too. Both solutions may have strongly parallel operation. It will be shown, that considerable parallelism can be realized in the case of calculations with the Hamiltonian model, too. Both solutions are based on simple "uniform" structures and modes of operation independent of the particular details of the concrete problem to be solved (min-max operations in the case of a fuzzy system and learning/operation of the ANN). The combination of the simple algorithms mentioned above as well as the use of the closed analytical formulas also independent of the particular details of the problem to be solved also gives similar advantages to the use of the Hamiltonian
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Model. Furthermore, for tuning the parameters within the Hamiltonian Model also gives room for combining it with different Soft Computing methods. The aim of this paper is to fully summarize the mathematical background of the proposed method in details, and to give illustrative examples regarding its operation via numerical simulations. It also is a goal to raise different ideas needing further investigations. In Section 2 the connection between the Lagrangian and the Hamiltonian models is brifly summarized since this step is the only link which connects the abstract Hamiltonian model to the phenomenologically well established and interpreted Lagrangian description, that is to the realm of industrial sensors. In Section 3 the advantages of the Hamiltonian model are summarized in a succinct way listing the basic analogies between the Euclidean and the Symplectic Geometry defined in the tangential space of the set of possible physical states of the system in the form of a table. Also in this section those main properties of the Lie Groups will be summarized which are systematically utilized in this paper in two different particular cases: in the case of the Orthogonal and the Symplectic Groups. This Section contains a table, too, in which certain common, most useful properties of Lie groups are summarized. Section 4 is devoted to the Deformation Principle on which the proposed conrol method is based. Section 5 is dedicated to certain ideas already raised in connection with the application of the Hamiltonian Model in control technology. In Section 6 certain particular considerations pertaining to a particular paradigm and the appropriate simulation results are presented to demonstrate the possibilities "hidden" in the Hamiltonian Model. The conclusions are drawn in Section 7, while the remaining sections contain those parts which are obligatory components of scientific publications. 2. The connection between the Lagrangian and the Hamiltonian Model As is well known, phenomenological foundations of CM were established by Galilei in the 16th century by realizing the role of time as an independent variable in describing the behavior of mechanical systems (Szamosi6) and by Newton in the 17th Century by introducing the concept of the Inertial Systems of Cordinates with recpect to which the
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behavior of a mass-point can be described in the simplest mathematical form. This description uses directly measurable physical quantities as coordinate-vectors, velocities, accelerations and forces for describing the equations of motion for a single mass-point. It is a simple mathematical consequence that by the use of an inertial frame the kinetic and potential energy of the system can be constructed for a set of mass points interacting with each other and with an external environment, and that the Newtonian equations of motion can be deduced from the energy function via simple mathematical operations, too. From this point of view rigid bodies can be considered as special objects for the full description of the motion of which the use of a few independent coordinates, the generalized coordinates in this paper consistently denoted by letter "q" can be used. On this basis it became possible to express the Newtonian equations of motion as a simple consequnce of a variational principle for conservative systems. This principle is called the Hamilton Principle running as follows (e.g. Arnold7): by using the system's Lagrangian as L(q, q ) = T V(q), T
1 M ij (q) q i q j , V(q) = potential energy 2
(1)
in which T denotes the kinetic energy, from the set of the system's prospective trajectories starting from point q0 in time t0 and ending in point q1 in time t1 Nature chooses the realized one for which the integral t1
L Q
Free u
q u dt Extremum
(2)
t0
In Eq.(2) there is a summation over the operative index denoted by subscript "u". The term QuFree denotes the generalized forces defined as Q iFree
qi
F x s t
s t
(q)
(3)
s
by using the Cartesian coordinates of the appropriate (sth) mass-points of the body with respect to the inertial frame xts. It can be shown, that in Eq.(3) for a conventional robot the components of QFree can be interpreted as the projection on the appropriate joint axis of the external forces/torques acting on the given arm section or link. At least in principle, the
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phenomenological basis of the Lagrangian model consists in this statement. By applying the usual method of partial integration the Euler-Lagrange equations of motion can be derived from the condition in Eq.(2) as d L L Q iFree . dt q i q i
(4)
Via applying the usual Legendre transformation the so called canonical momentum "p" and the Hamilton function (Hamiltonian) H(p,q) can be introduced as pi
L M ij (q) q j , H(q, p) p s q s L . q i
(5)
Via introducing L into Eq.(2) the variational principle yields the equation of motion in the terms of the canonical coordinates as p s
H H Q sFree ,q s . qs ps
(6)
From practical point of view it is worthy of note, that the canonical momentum "p" normally does not have directly measurable components. It is related to the directly measurable ones by the elements of the inertia matrix Mij which contain the inertia parameters of the robot arm and the gripped work-piece manipulated by the robot. In the practice normally no accurate informmation is available on the values of these parameters. Though the same parameters also are present in the Lagrangian, at least the independent variables of the system within the Lagrangian model are directly measurable in principle+. This fact may explain why the Hamiltonian formalism is almost completely "neglected" in connection with the control of conventional robots in the present literature. Though from phenomenological aspect the Hamiltonian model seems to be quite disadvantageous, regarding its mathematical structure it leads to appropriate equations of +It
has to be noted, that in the practice the situation is not so clear even in the case of the Lagrangian model. In the definition given in Eq.(3) the Fts components should be summed over each "elementary" mass point of the robot arm as a rigid body. However, this summation could be done only if the whole surface of the robot would be covered with local force sensors. It is only a special supposition that the point of the action of the external forces is located in the gripper and that this external interaction can completely be identified via force and momentum sensors.
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motion of far simpler structure that of the Lagrangian model. The main advantage of this model is that both the generalized coordinaes and the canonical momentums have "equal rights" within the set of the first order differential equations characteristic to the Hamiltonian concept. This simplicity has profound mathematical consequences still poorely utilized in robot control. These consequences are considered in the next Section. 3. The formal advantages of the Hamiltonian model By "putting together" the components of "q" and "p" in a 2 DOF dimensional array defined as xT=[qT,pT], the full energy of the system can be expressed as a simple scalar function H(x). By introducing the constant skew symmetric matrix of unit determinant 0 I
- I 0 ~
T
and the "2 DOF" dimensional array Q Free 0 T , Q Free
(7)
T
for the external generalized forces the
equations of motion gain the form of x i ij
H x ~ Free Qi . x j
(8)
Eq.(8) gives rise to a very simple geometric interpretation. Let us suppose, that the appropriate "x" coordinates can be the elements of some open region . In each point "x"of x x ( n ) the tangential space of the states can be defined as the linear space of the (n) x x
vectors in which x(n) runs over the set of the neighboring points within the infinitesimally close vicinity of x. x evidently is an element of this set. The possible states of the system can be considered as a differentiable manifold for the mathematical description of which atlases consisting of consistent (contradiction-free) maps are used. A particular map realises a mapping of a sub-set of the manifold onto an open sub-set of 2 DOF. Due to the considerations related to the existence of the inertial frames it can be stated, that Nature distinguishes a special map the coordinates of which correspond to those canonical variables which directly can be introduced from the Lagrangian generalized coordinates.
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From this point on a higher level of abstraction can be achieved in describing CM by turning from this phenomenologically well substantiated "Lagrangian Map" to other maps. In the tangential space of the system's states this immediately leads to algebraic and geometric analogies as it is shown below. 3.1. Analogies between the Euclidean and the Symplectic Geometries From purely mathematical aspect other maps can be introduced by an arbitrary differentiable and invertable time-independent coordinate transformation x'(x) treating the Hamiltonian as an invariant scalar, that is changing its form as function as H'(x'(x))H(x). The equations of motion according to this new map can be deduced from the original ones as x i
xi x H ~ Free xi H xu ~ Free H ~ Free st Qs Qs Aiu x Qi x s i st x s x s x t xu xt xu x s
(9)
It is clear, that the structure of Eq.(9) is very similar to that of Eq.(8). In this case a skew symmetric nonsingular and non-constant matrix A(x) stands for , and a similar, 2 DOF array stands for the external generalized forces. Both of them obeys well defined transformation rules as given in Eq.(9). The situation is quite similar to the transformation law of the matrix elements of the metric tensor in the case of an Euclidean Geometry when turning to new coordinates defined with curved surfaces. In this case the coordinate depenence in "A" does not convey any essential information on the physical system for the description of which it is used. It is rather characteristic to the more or less arbitrary way according to which the coordinates of a new map can be chosen. As in the case of the Euclidean Geometry, in which the possibility for introducing special coordinates leading to the constant unit matrix as the representation of the metric tensor distinguishes these systems as particular ones yielding the simplest description, in CM the possibility for choosing special maps on which A(x) const. also distinguishes these maps leading to the possible simplest form of the equations of motion. Therefore, as in the case of Euclidean Geometry of particular interest are those
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transformations which leave the form of the metric tensor unchanged, in CM those coordinate transformations of the form of x ( x ) for which xi x st u iu or in matrix form SST= x s x t
(10)
also have special significance. (Here the matrix "S" stands for the Jacobi matrix of the coordinate transformation). From Eq.(10) it is clear, that detS= 1. With the restriction of detS=1 these transformations are referred to as Canonical Transformations, and the appropriate maps are called Canonical Maps. The appropriate Jacobians S(x) are referred to as symplectic matrices. From this point on it is easy to summarize tha main formal analogies between the Euclidean and the symplectic geometries in Table I. Table I: The formal analogies between the Euclidean and the Symplectic Geometries 3.2. Other advantages of the Hamilonian formalism As normally in different fields of Classical Physics the basic laws of nature can be expressed in tensorial form based on the structure of the scalar product, within the frames of Classical Mechanics the symplectic structure has similar distinguished significance. Any measurable physical quantity characteristic to the system must be an unique function of the canonical coordinates unambiguously describing its physical state. The evolution of such a quantity f(x) for an autonomous system can be described as a "Poisson Bracket" defined on the basis of the symplectic structure as f f H f (x) x i ij { f , H} . x i x i x j
(11)
It is clear, that for two arbitrary scalar functions f(x) and g(x) the x'(x) canonical transformations lead to the representation f'(x'(x))f(x), etc. leaving both the numerical value and the form of Eq. (11) unchanged.
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From algebraic point of view the linear space of the arbitrary many times continuouisly differentiable functions can be transformad into an algebra by considering the Poisson brackets as a multiplication. Skew symmetry of the matrix has two significant consequences which can be utilized in the parctice. They are as follows: a)
If for a function f(x) {f,H}0, from this it follows, that {H,f}{f,H}=0. This fact can be interpreted as a symmetry principle: if the phase current generated by f(x) as x i ij
f leaves the Hamiltonian of the system unchanged (that is it is a symmetry of x j
the system), than the evolution of the system's state defined by the phase current x j js
H leaves the numerical value of "f" constant. That is, to each symmetry of x s
the system pertains a characteristic constant. b)
For three arbitrary, infinitely many times continuously differentiable function f(x), g(x) and h(x) this algebra has the properties of a Lie algebra, that is the Jacobi identity is satisfied by them: {f,{g,h}}+{g,{h,f}}+{h,{f,g}}0. If we put the Hamiltonian of the system into the place of h(x), and f(x) and g(x) pertain to some symmetries of the system, their Poisson bracket {f,g} will also be a symmetry of the same system.
Since the motion of the system is determined by the 2DOF constants determining the initial conditions, finding some symmetries will help us to solve the equations of motion. By the systematic use of the Poisson bracket new constant quantities can be constructed from the known ones. 3.3. Common aspects of the Orthogonal and the Symplectic Groups utilized Lie groups are special groups the elements of which can be "parametrized" by continuous parameters in the form of g(!) and g(") in a way that the product of the elements g(#)=g(!)g(") the is a unique function of the parameters #(!,") and it is continuously differentiable in infinite times. (!, ", # are the elements of N, in which N denotes the dimension of the parameter-space of the group. If "=0 corresponds to the unit element of the 13
group, in an arbitrary composite scalar function of the variable "t" g("(t)) for which "(0)=0 the d g (t ) G dt
(12)
quantities can be considered as the elements of the tangential space of the group drawn at the unit element. For the generators of a Lie group simple considerations can be done leading to important consequences as is given in Table II. Its is evident, that for an arbitrary generator G the function defined by the power series of the matrix exponential g(t)=exp(tG) represents a single-parameter sub-group generated by G. From the finite dimension of the linear space of the generators it immediately can be concluded, that by using the appropriate number of linearly independent parameters G(i) the matrix product g ( p1 ,..., p N ) exp p1G (1) exp p 2 G ( 2 ) ... exp p N G ( N )
(13)
yields a special continuous parametrization of the group. Table II: Certain common properties of Lie groups utilized in the adaptive control
Normally by using special linearly independent generators the power series of the matrix exponentials can easily by expressed in a simple closed analytical form. It also is worthy of note, that if the vector v is an eigenvalue of the generator "G", then for an arbitrary element of the group "g" the gv vector will be the eigenvalue of the generator gGg-1,, since if Gv=$v, then gGg-1gv=$gv.
(13)
To a special case corresponds the $=0 to which the vectors left unchanged by the exponential in exp(tG) pertain. It is trivial, that if for a given G the matrix exponential has a closed form, than the matrix exponential for an arbitrary group element "g" can also be expressed in closed form, since exptg Gg 1 g exptG g 1 .
(13)
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For gaining the generators of a Lie group the near unity transformations can be considered which must have the form of T = I + % A, in which % is a small real number. This can be substituted into the quadratic equation defining the group. By prescribing the fulfillment of the definition up to the first power of % the appropriate restrictions for the structure of the possible generators can be gained. In the case of the orthogonal group the generators must be skew symmetric matrices. For the symplectic group they must have the form of G a , b A a ,a G T
G b ,b T
G a ,b
(14)
T
in which Ga ,a Ga ,a , G b,b G b,b otherwise are arbitrary matrices of DOF DOF dimensions and Ga,b is arbitrary. It is trivial, that AT has similar structure, that is if A is generator, then AT also is a generator. Since the generators form a linear space, symmetric and skew symmetric generators can be introduced for the symplectic group as K
H
H J
, K H
H
J
(15)
where H+, H- and K are symmetric, and J is skew-symmetric. By using these block matrices as linearly independent generators the closed analytical formulas for the exponentials are given below. K 0 exp(tK ) 0 exp t
tK ) 0 K 0 exp( J 0 exp(tJ ) 0 exp t
0 exp(tJ ) 0 J
(16)
0 H ch tH sh tH exp t
H 0 sh tH ch tH 0 H costH sin tH exp t
H 0 sin tH costH
For our control technical aims construction of symplectic group generators leaving certain vectors u unchanged will be necessary. Such generators can easily be found by decomposing u into two blocks as u=[aT,bT]T, and by prescribing the restrictions for te symmentric and the skew symmetric blocks of the generators as
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Ka + H- b = 0, H- a Kb = 0, Ja + H+ b = 0, H+ a +Jb = 0.
(17)
These restrictions can automatically be satisfied if the matrices J, H-, and H+ are constructed of arbitrary vectors taken from the common orthogonal sub-space of a and b. Since the "a parallel with b case" may occur very seldom, generally it may be supposed, that this subspace is of the dimension of (DOF2) spanned by the linearly independent orthonormed basis vectors {c(i)|i=3,...,DOF}. The appropriate components in the generators are
K ij( uv ) H ij( uv ) H ij( uv ) J ijuv
1 (u ) (v ) ci c j c (ju ) ci(v ) , 2 1 ci( u ) c (jv ) c (ju ) ci( v ) 2
(18)
Such a set can easily be created by the use of the Gram-Schmidt algorithm so started with an initially orthonormed set that its first two elements are replaced by a and b. The columns of these unit vectors form an orthogonal matrix C which can be utilized in the control for assigning continuous parameters to certain symplectic matrices for the purpose of parametertuning. 4. The "Deformation Principle" In strict analogy with the idea invented by L. Jánossy, in the field of the Hamiltonian Mechanics introduction of the following principle can be attempted. Within the frames of the conventional theory the canonical x'(x) transformations allowed only one way for interpretation: "the same physical system is described by the coordinates of a new canonical map". The novel interpretation of certain canonical transformations: "the x'(x) coordinates correspond to an other, hypothetical physical system (the 'deformed one') described by the use of the coordinates of the original map". In order to use this principle for control-technical purposes it is necessary to use the directly measurable quantity y q, q instead of the canonical x. The question is how these quantities can be related to the canonical formalism of Classical Mechanics. By the use of the functions
16
H y H xy , L
~ x ~ Free , Q LQ Free a skew-symmetric, non-singular matrix A(y) can be y
introduced instead of as Ay L1 L1 . The equations of motion gain a form more or less T
similar to the canonical ones: y i Aij y
H y ~ Free Q . Instead the original canonical y j
transformations modified ones of the form of z(y), for which U
&y leaves A(y) locally &z
unchanged can be considered: Az (y ) U 1 Ay U 1 Ay . If we apply a very rough T
approximate dynamic model for the system, the matrix L can be constant, consequently the matrix A can also be constant. It is evident that the above defined U matrices form a group, too, and this group is in quite close realtion with the symplectic group and have very similar properties: the matrix LU-1L-1 matrices are symplectic. The applied control method is based on this idea. The appropriate restrictions to be imposed for the purposes of the deformation principle is as follows: In the canonical map directly deduced from the Lagrangian model the generalized force vectors have only DOF non-zero components. Since a general canonical transformation can "mix together" all the 2DOF components of the transformed generalized force vectors, a considerable part of the canonical transformations cannot be applied for deformation purposes. Only those solutions can be accepted, for which the necessarily "truncated", phenomenologically non-interpretable components of the generalized force vector are negligible in comaprison with the interpretable parts. 5. Possible applications of the deformation principle in adaptive control The essence of the deformation consists in the difference in the phase currents generated by H' and H in the same point of the differentiable manifold. The idea of partial system identification is related to this interpretation: starting with a very rough initial model of constant M in L and with a constant &V/&q the model establishes a connection between the exerted local generalised forces and the propagation of the state-vector y Mod . In the reality the encoders measure a different propagation y Re al ' y Mod . It is expected, that the difference can be eliminated by some deformation of the initial model in the form of H'(z)=H(y(z)) locally represented by the appropriate matrix U.
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According to the original canonical formulation, an appropriate symplectic matrix is to be found for which a x Re al Sx Mod Sb is valid. This can be done e.g. in the following way: By making two quadratic matrices of the column vectors a and b (A and B) via "putting near them" further linearly independent vectors the matrix relation A=SB can be prescribed. Due to the group properties of the symplectic matrices this can be satisfied if both A and B are symplectic and their first column is equal to a and b respectively. The solution is simply S=AB-1=ABTT. This situation can be achieved by the symplectising algorithm, a simple and easily programmable procedure quite similar to the Gram-Schmidt orthogonalization method frequently used in Qantum Mechanics for gaining orthonormed basis vectors. The two algorithms can be treated in a strictly "parallel" way as it is summarized in Table III. Due to the group properties of the symplectic matrices in each computing cycle of the controller the symplectic deformation applied in step "t" S(t) can be so "corrected" by the actually computed symplectic correction T(t) that S(t+1)=T(t)S(t), etc. It can be expected, that this modified model will yield better solution than the original, "rough" dynamic model without any corrections. An alternative possibility is to refuse the idea of cumulative corrections and applying symplectic deformation in each steps starting directly from the initial rough dynamical model in each step. It is evident, that in both cases the symplectic model based approach contains a considerable number of "unconstrained parameters" hidden in the columns of matrices A, B. Though the symplectising algorithm decreases the number of these parameters, within this process the "story" of these parameters cannot be traced in a lucid way. Furthermore, though these parameters do not concern the control task in the given step, the appropriate prediction made on the basis of this estimation influences the behavior of the controlled system in the next step.
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In order to deal with the free parameters in a more flexible way, introduction of the continuous Lie parameters is expedient. It may be done in the following way. Instead using its slight modification can be introduced in the form as (19)
P* SDesired = S P SMeasured
in which P* and P are independent matrices containing the continuous parameters of the symplectic group, and they are so chosen, that P*uDesired=uDesired, and PuMeasured=uMeasured. According to the procedure based on the Gram-Schmidt algorithm and Eq. (18) an orthogonal matrix C can be constructed in each control step. Due to the group properties of the orthogonal matrices it can be considered as a rotation of an "initial set" forming the columns of the unit matrix as c(i) = C e(i), e(i)j = (ij.
(19)
By applying Eqs. (13-14) for the orthogonal group, the appropriate blocks K, J, H-, H+ constructed of the {c(i)} vectors and their analytical functions can simply be constructed from their counterparts constructed of the set {e(i)}: K=CK'CT,, J=CJ'CT, H+-=CH+-'CT. These expressions can be applied in the blocks of Eq. (16). Since the components of the {e(i)} set have very simple structure, their appropriate matrix exponentials for a 3DOF system can be expressed in closed analytical form as
S K 33
1
0
0
0
0
0
0 0 1 0 0 exp(t ) 0 0 0 0 0 0
0 0 0 1 0 0
0 0 0 1 0
0 1 0 0
0
0 0 H 33 0 0 ch(t )
, S 0 0
0
0 0
0 0 1 0
0
0 exp(t )
0 0 sh(t )
0 0 0 1 0 0
0 0 0 1 0
0 1 0 0
0
0 sh(t ) H 33 0 0 cos(t )
, S 0 0
0
0 0
0 0 1 0
0
0 ch(t )
0 0 sin(t )
0 0 0 1 0 0
0 0 0 0
0 sin(t )
, 0 0
1 0
0 cos(t )
(In the case of 3 DOF systems for the skew-symmetric J only the trivial zero solution can be gained.) The matrix denoted by the superscript "K" expresses a stretch/shrink along the third axis, "accompanied" with a simultaneous contraction/stretch in the third component of the canonical momentum vector. The matrix that has the superscript H- describes a hyperbolic rotation in the space of the canonical state vectors influencing only the third components of the state vectors. Similar statement holds for the matrix denoted by H+ describing "common rotation". Via associating continuous parameters )1, )2, )3 to K, H- and H+ in the symplectic
19
matrix P, and assigning their counterparts *1, *2, and *3 to P*, the control of a 3DOF system development of parameter-tuning strategies can be initiated. In the next section the behaviors of certain strategies are presented on the basis of computer simulation. 6. Simulation results For simulation purposes exactly the same 3DOF robot arm srtructure was used that in a previous investigation8. It consisted of a vertical rod of 5 kg moving up and down (q1 in m), rotating around itself as a vertical axis (q2 in rad), and a second rod joined to it by a wrist tilting around a horizontal axis (q3 in rad). This latter joint also was translated by q1 and rotated by q2. The second rod had negligible mass but carried a point-like small body of variable mass. It also had variable, but constant length (R0 in m). The three axes were controlled by drives exerting force for q1 and torque for q2 and q3 prescribed by the control strategy. In each cases considered the end-point of the robot arm was desired to be moved with circular frequency s-1 along a circle of 0.5 m radius laying in a vertical plane at a distance of 2 m from the vertical axis. In each cases the "initial rough estimation" of the dynamic model consisted of a non-singular, constant inertia matrix and a constant gravitational term. No quadratic velocity coupling was taken into account. Making all the further corrections was the task of the symplectising algorithm and the "predictive control". In the space of the joint coordinates a linear feedback was applied as D q N bq R q N cq R q N q
(19)
in which the superscript D, R and N correspond to the "desired", the "realized" and the "nominal" values, respectively. (According to the above concepts, there is no "desired" trajectory. The nominal trajectory pertains to the motion to be executed. In the case of a Computed Torque Control (CTC) we have only desired accelerations in each control cycle.) The unmodelled environmental interaction was represented by a spring having viscous friction ("dashpot"). The values b=15 s-1 and c=56 s-2 were so chosen that Eq.(19) approximately corresponds to a single damping constant without any oscillation. In the lack of the exact dynamical model this strategy cannot be precisely implemented. The expected role 20
of the identification method was continuous correction of the initial dynamic model in order to really implement this strategy. In the graphs the solid, the dashed and the dashdot lines correspond to q1 (Q1), q2 (Q2), and q3 (Q3), respectively. (In the figures describing functions versus time time is given in (t units, that is the duration of one computational cycle. In the simulations it was 5 ms.) In the first quarter of the duration of the motion only a simple linear feedback based on the rough dynamical model was applied. The symplectic model deformation was in effect only from this point on. Regarding the tuning of the continuous Lie parameters, in each cases a very simple strategy was alpplied: the adaptive parameters were kept moving. If the change in the adaptive parameter coincided with the decrease in the prediction error in the state propagation, this tendency was maintained. Otherwsie it was reversed. Application of a non-cumulative approach in which the input of the symplectising algorithm were the columns of the symplectic matric T did not give satisfactory results as it is given in Fig. 1.
Fig. 1: Typical result of the non-cumulative solution without tuning of the adaptive parameters projections in the planes of the pahse-space and the joint coordinate errors Its quality was not better than that of the purely linear approach. Tuning of the adaptive parameter )1 (the lie-parameter of the stretch/contraction along e(3)) did not give essential improvement as it is shown in Figs. 2-3. Fig. 2: Typical result of the non-cumulative solution with tuning )1 : projections in the planes of the pahse-space and the joint coordinate errors Fig. 3: Variation of )1 for parameter tuning. Increasing or decreasing the finite steps in the adaptive parameter did not give essential modification of the above sructure. Consequently, the further investigations were concentrated on the behavior of the cumulative approach.
21
Regarding the cumulative approach, the symplectizing algorithm with the same inputs showed better results but it was very sensitive to the viscosity present in the "dashpot" as in the counterpart in an unmodelled external interaction. A similar solution using the columns of the unit matrix as the input of the symplectizing algorithm resulted in a better quality of motion even without extra parameter tuning. The results are described in Figs. 4a-4b.
Fig. 4a: Cumulative control without parameter tuning for small viscosity: phase space, joint coordinate erorrs; Fig. 4b: Cumulative control without parameter tuning for small viscosity: phenomenology test (the norm of the truncated, non-interpretable part of the generalized forces.) and the generalized forces Figs. 4a-4b reveals, that the cumulative nature of the control by itself means a kind of adaptivity. The applied transformations satisfy the deformation principle since the truncated part of the generalized forces is negligible in this case. Due to its cumulative nature this approach is much more sensitive to the variation of the continuous parameters as it is shown in Figs. 5-7: The modification is mostly apparent in the figures describing the joint coordinate errors.
Fig. 5a: Behavior of the cumulative control when tuning )1 and the viscosity is small: phases space, joint coordinate errors (dimless). Fig. 5b: Behavior of the cumulative control when tuning )1 and the viscosity is small: phenomenology test and )1 in the range of 10-3. The system is far more sensitive to the variation of )2 and )3 "mixing" the phenomenologically interpretable and non-interpretable components. Fig. 6a: Behavior of the cumulative control when tuning )2 and the viscosity is small: phase space, joint coordinate errors
22
Fig. 6b: Behavior of the cumulative control when tuning )2 and the viscosity is small: phenomenology test, and )2 in the range of 10-6.
Fig. 7a: Behavior of the cumulative control when tuning )3 and the viscosity is small: phase space, joint coordinate errors Fig. 7b: Behavior of the cumulative control when tuning )3 and the viscosity is small: phenomenology test, and )3 in the range of 10-5. In Figs. 8a-8b aech of the six continuous parameters are tuned during 3-3 consecutive steps in the case of low viscosity. In the case of the tuning applied the interdependence of these parameters shows a kind of stabilizing effect: none of them can meander far from the zero value pertaining to the identical transformation.
Fig. 8a: Tuning each of the continuous parameters in the case of small viscosity: phase-space, joint coordinate errors Fig. 8b: Tuning each of the continuous parameters in the case of small viscosity: phenomenology test, generalized forces to be exerted by the drives In Figure 9 the effect of a high viscosity coefficient can be traced. It can be stated, that the given strategy is not too sensitive to the viscosity of the external system (dashpot):
Fig. 9: The effect of a greater viscosity in the environmental coupled system: the phasespace, joint coordinate errors The effect of the increased viscosity can well be observed in thes phase space, in the shift of the joint coordinate errors and in the change in the shape of the curves describing the generalized forces. The data described in Fig. 10a-10b pertain to very high environmental viscosity. The above mentioned tendencies are far more easily observable in these graphs:
Fig. 10a: The effect of very high viscosity in the environmental coupled system: the phasespace, joint coordinate errors
23
Fig. 10b: The effect of very high viscosity in the environmental coupled system: phenomenology test and the generalized forces
7. Conclusions In this paper a concise theory has been developed with the aim of making it possible to utilize certain simple concepts inherent in the Hamiltonian Mechanics for control technical purposes. The theoretical approach here presented was strongly based on simple formal analogies between otherwise different mathematical concepts and procedures. It was found, that on the basis of an idea entitled as "deformation principle" the transparent concepts as canonical transformations, the Euclidean and the Symplectic Geometry, the Orthogonal and the Symplectic Groups as Lie groups and their generators, the orthonormed and the symplectic sets of basis vectors, the Gram-Schmidt and the Symplectizing algorithms can be used for inventing robust and adaptive control for mechanical systems in dynamic interaction with an unmodelled environment via tuning of continuous free parameters within closed form analytical expressions. The number of the independent continuous, tunable parameters strongly increases with the degree of freedom of the mechanical system to be controlled. Neither the complexity, nor the structure of the computational operations depend on the particular features of the mechanical system to be controlled. This structure has a kind of "uniformity" and universality as certain ANNs and fuzzy controllers has. The proposed algorithms can be runned in a strongly parallel way on an appropriate, multiple-processor hardware: for both sides of the control equation computation of the change in the Lieparameters, the Gram-Schmidt and the Symplectizing algorithms can be runned simultaneously in a parallel way. It was also found, that from a well defined point on a great variety of the possible tuning strategies can be developed. By the use of a particular paradigm and computer simulations two typical versions were investigated: the non-cumulative and the cumulative approaches. Comparison of the results revealed, that the cumulative approach seems to be far more
24
effective that the non-cumulative one. The applied simple heuristic tuning strategy based on consecutive tuning of the independent continuous parameters according to the results of simple correlation-investigations concerning the accuracy of the prediction of the motion shows stability near the unit transformation. This strategy was found to be less sensitive to the viscous interactions than its more heuristic progenitors starting the symplectizing algorithm from T instead of I. It is worthy of note, too, that the control strategy applied also contains important parameters which do not form the part of the symplectic model. The possible effects of these parameters were not investigated in this paper. It is likely that making further investigations in connection with different paradigms and parameters will be reasonable. 8. Acknowledgment The authors gratefully acknowledge the grant provided by the "Fund for the Hungarian Higher Education" (AMFK'95) and that of the "National Scientific Research Fund" (OTKA T019032)
9. References 1
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2
Tosunoglu, S. Fault Tolerant Control of Mechanical Systems. In proc. of IEEE 21st International Conference on Industrial Electronics (IECON'95), 6-10 November 1995,Orlando, Florida. Vol. 1, pp. 110-115.
3
B. Lantos, "Identification and adaptive control of robots", International Journal Mechatronics, Vol. 2, No. 3, pp. 149-166, (1993). 25
4
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5
J.K. Tar, O.M. Kaynak, J.F. Bitó, I.J. Rudas, D. Mester: "A New Method for Modelling the Dynamic Robot-Environment Interaction Based on the Generalization of the Canonical Formalism of Classical Mechanics", accepted for publication in the "First International ECPD Conference", Athens, Greece, July, 1995.
6
G. Szamosi: "Polyphonic Music and Classical Physics, the Origin of the Newtonian Concept of Time", History of Science, 28. pp. 175-191 (1990).
7
V.I. ARNOLD: "Mathematical Methods of Classical Mechanics" (original issue in Russian by "Nauka"), Hungarian translation issued by Mûszaki Könyvkiadó Budapest, Hungary 1985.
8
I.J. Rudas, J.F. Bitó, J.K. Tar: "An Advanced Robot Control Scheme Using ANN and Fuzzy Theory Based Solutions", Robotica, Vol. 14, pp. 189-198 (1996).
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